The local integrability condition for wavelet frames

In this article we study finitely generated wavelet systems with arbitrary dilation sets. In 2002 Hernández et al. gave a characterization of when such a system forms a Parseval frame, assuming that a certain hypothesis known as the local integrability condition (LIC) holds. We show that, under some mild regularity assumption on the wavelets, the LIC is solely a density condition on the dilation sets. Using this new interpretation of the LIC, we further discuss when the characterization result holds.